Existence of a solution; Initial value problem

Hello

Can somebody help me to solve this problem:
Given an ODE $\displaystyle x'=f(t,x)$ continous and lokally lipschitz for x. $\displaystyle x(t), t \geq t_0$ is a solution to the initial value problem $\displaystyle x(t_0)=x_0$. $\displaystyle x_1(t) $ and $\displaystyle x_2(t)$ are differentiable and: